# Kunihiko Kodaira

# Kunihiko Kodaira

**1915-1997**

**Japanese Mathematician**

In a long career on two continents, Kunihiko Kodaira conducted ground-breaking research in the areas of algebraic varieties, harmonic integrals, and complex manifolds. He also became the first Japanese recipient of the Fields Medal in 1954, and he authored a number of textbooks that served to increase understanding of mathematics among Japanese students.

Born on March 16, 1915, in Tokyo, Kodaira was the son of Gonichi and Ichi Kanai Kodaira. He lived in Japan's capital during his early years, including his education at the University of Tokyo, which began when he was 20. There he studied a variety of mathematical fields and published his first paper (written in German) a year before he graduated in 1938. Among his early influences were John von Neumann (1903-1957), André Weil (1906- ), Hermann Weyl (1885-1955), M. H. Stone, and W. V. D. Hodge. Also significant was the influence of the book *Algebraic Surfaces* by the Italian geometer Oscar Zariski, which sparked Kodaira's interest in algebraic geometry.

Kodaira continued his university education, earning a second degree in theoretical physics in 1941. By this point, of course, Japan was at war with many of the world's powers, and this left Japanese mathematicians isolated from the research taking place in the university centers of Britain, the United States, and other Allied nations. Yet he managed to investigate a number of challenging problems, and during the war years formed many of the ideas that would undergird his doctoral thesis. In 1943 Kodaira married Sei Iyanaga, and they later had four children.

He received an appointment as associate professor of mathematics at the University of Tokyo in 1944, and continued working on his thesis, which concerned the relation of harmonic fields to Riemann manifolds. A harmonic entity in mathematics is something whose interior can be described simply by examining only the boundary of its surface, and Weyl had been noted for his research in Riemann surfaces, a realm of topology associated with the non-Euclidean geometry of G. F. B. Riemann (1826-1866). Now Kodaira sought to establish the concept of a Riemann manifold, a type of topological space.

Just after Kodaira received his doctorate in 1949, Weyl read his dissertation in an international mathematics journal, and was so impressed that he invited the young Japanese mathematician to join him at the Institute for Advanced Studies (IAS) in Princeton, New Jersey. Kodaira accepted the invitation, and moved his family to the United States, where they would remain for the next 18 years.

During his time at Princeton, Kodaira had an opportunity to work with some of the world's leading mathematicians, including W. L. Chow, F. E. P. Hirzenbruch, and D. C. Spencer. In the course of his research there, he discovered that by use of harmonic integrals, he could more completely define Riemann manifolds. Complex manifolds—that is, manifolds involving complex numbers—are essential to the study of modern calculus, but at that time the properties of many such manifolds were undefined; hence Kodaira's work was highly significant.

Kodaira also investigated Köhlerian manifolds, which he sought to prove were analytic in nature, just like Riemann manifolds—i.e., that they could be solved or defined by calculus. This led him into investigation of a Köhlerian subset, Hodge manifolds. Using what he called "the vanishing theorem," an idea he had developed, Kodaira was able to prove that Hodge manifolds were analytic. He then proved, by using another theorem he had conceived, the "embedding theorem," that this in turn meant that all Köhlerian manifolds were analytic as well.

Needless to say, this was all highly specialized work at the furthest cutting edge of contemporary mathematics, and given the great differences between various types of manifolds, Kodaira's contributions to their classification was a significant one. For this he received the Fields Medal in Amsterdam in 1954, with his mentor Weyl pinning the medal on him. His native country later awarded him two of its highest honors, the Japan Academy Prize and the Cultural Medal.

Kodaira spent a year at Harvard in 1961, then received an appointment to Johns Hopkins University. After three years at Johns Hopkins, he accepted the chair of mathematics at Stanford University, but in 1967 he returned to the University of Tokyo. There he began writing a number of works, including a series of textbooks commissioned by the Japanese government. He died in Kofu, Japan, on July 26, 1997.

**JUDSON KNIGHT**

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